While Iqbal Khan and Khurshid [20] gave several Hermite-Hadamard type in- equalities for convex functions via generalized fractional integrals

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https://doi.org/10.5831/HMJ.2020.42.4.653

FRACTIONAL INEQUALITIES FOR SOME EXPONENTIALLY CONVEX FUNCTIONS

Naila Mehreen^∗ and Matloob Anwar

Abstract. In this paper, we establish new integral inequalities via Riemann-Liouville fractional integrals and Katugampola fractional integrals for the class of functions whose derivatives in absolute value are exponentially convex functions and exponentially s-convex functions in the second sense.

1. Introduction

Study of generalized convex functions is important due to its signif- icant application in integral inequalities. On the other hand, fractional integrals also play role in the advancement of integral inequalities.

The Hermite-Hadamard inequality [9, 8] for a convex function F : H → R on an interval H is defined by

(1) F h₁+ h2

2

≤ 1

h2− h₁ Z h2

h1

F (g)dg ≤ F (h₁) + F (h2)

2 ,

for all h₁, h₂ ∈ H with h₁ < h₂. Inequality (1) is then proved for other generalized convex functions, for instance Du [7], Khan [14, 15] and Khurshid [19] proved several inequalities for generalized convex functions. Also see [1, 3, 5, 6, 24, 23]. While Iqbal [10, 11, 12], Khan [16, 17, 18] and Khurshid [20] gave several Hermite-Hadamard type inequalities for convex functions via generalized fractional integrals.

Awan et al. [1] introduced following new class of convex functions.

Received January 14, 2020. Revised August 19, 2020. Accepted August 25, 2020.

2010 Mathematics Subject Classification. 26A51, 26D07, 26D10, 26D15.

Key words and phrases. exponentially convex function, exponentially s-convex function, Riemann-Liouville fractional integrals, Katugampola fractional integrals.

The present investigation is supported by the National University of Sciences and Technology (NUST), Islamabad, Pakistan..

*Corresponding author

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Definition 1.1 ([1]). A function F : H ⊆ R → R is called exponentially convex, if

(2) F (uh₁+ (1 − u)h₂) ≤ uF (h₁)

e^βh¹ + (1 − u)F (h₂) e^βh² ,

for all h1, h2 ∈ H, u ∈ [0, 1] and β ∈ R. If the inequality (2) is in reversed order then F is called exponentially concave.

Mehreen and Anwar [24] introduced another class of functions called exponentially s-convex in second sense.

Definition 1.2 ([24]). Let s ∈ (0, 1] and H ⊂ R0 be an interval. A function F : H → R is called exponentially s-convex in the second sense, if

(3) F (uh₁+ (1 − u)h₂) ≤ u^sF (h₁)

e^βh¹ + (1 − u)^sF (h₂) e^βh² ,

for all h₁, h₂ ∈ H, u ∈ [0, 1] and β ∈ R. If (3) is in reversed order then F is called exponentially s-concave.

Example 1.3. A function F : [0, ∞) → R, defined by F(g) = ln(g) for s ∈ (0, 1) is an exponentially s-convex, for all β ≤ −1.

Definition 1.4 ([21]). Let F ∈ L[h₁, h₂]. The right-hand side and left-hand side Riemann- Liouville fractional integrals J_h^α₁₊F and J_h^α

2−F of order α > 0 with h₂ > h₁≥ 0 are defined by

J_h^α₁₊F (g) = 1 Γ(α)

Z g h1

(g − t)^α−1F (t)dt, g > h₁, and

J_h^α₂₋F (g) = 1 Γ(α)

Z h2

g

(t − g)^α−1F (t)dt, g < h₂,

respectively, where Γ(·) is the Gamma function defined by Γ(α) = R∞

0 e^−tt^α−1dt.

In [25] and [26], authors proved the following identity via Riemann- Liouville fractional integrals.

Lemma 1.5 ([25]). Consider a differentiable mapping F : [h₁, h₂] ⊂ R+→ R on (h1, h2) with h1 < h2. Then the following equality holds:

(4) F (h₁) + F (h2)

2 − Γ(α + 1)

2(h2− h₁)^αJ_h^α

1+F (h^ρ₂) + J_h^α₂₋F (h₁)

= h2− h₁ 2

Z 1 0

[(1 − u)^α− u^α] F⁰(uh1+ (1 − u)h2)du.

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Definition 1.6 ([13]). Let [h1, h2] ⊂ R be a finite interval. Then, the left- and right-side Katugampola fractional integrals of order α(> 0) of F ∈ Xv^p(h1, h2) are defined by,

ρI_h^α₁₊F (g) = ρ^1−α Γ(α)

Z g h1

(g^ρ− t^ρ)^α−1t^ρ−1F (t)dt, and

ρI_h^α₂₋F (g) = ρ^1−α Γ(α)

Z h2

g

(t^ρ− g^ρ)^α−1t^ρ−1F (t)dt,

with h₁ < g < h₂ and ρ > 0. Where X_v^p(h₁, h₂) (v ∈ R, 1 ≤ p ≤ ∞ ) is the space of those complex valued Lebesgue measurable functions F on [h₁, h₂] for which kF k_X_v^p < ∞, where the norm is defined by,

kF k_X^p

v =

Z h2

h1

|t^vF (t)|^pdt t

1/p

< ∞, for 1 ≤ p < ∞, u ∈ R and for the case p = ∞,

kF k_X^∞

v = ess sup_h₁≤t≤h₂[t^u|F (t)|].

Where ess sup stands for essential supremum.

Chen and Katugampola [2] proved following generalized version of Lemma 1.5.

Lemma 1.7 ([2]). Let α > 0 and ρ > 0. Consider a differentiable mapping F : [h^ρ₁, h^ρ₂] ⊂ R+→ R on (h^ρ₁, h^ρ₂) with 0 ≤ h1 < h2. Then the following equality holds if the fractional integrals exist:

(5) F (h^ρ₁) + F (h^ρ₂)

2 −αρ^αΓ(α + 1) 2(h^ρ₂− h^ρ₁)^α

_ρ

I_h^α₁₊F (h^ρ₂) +^ρI_h^α₂₋F (h^ρ₁)

= h^ρ₂− h^ρ₁ 2

Z 1 0

[(1 − u^ρ)^α− (u^ρ)^α] u^ρ−1F⁰(u^ρh^ρ₁+ (1 − u^ρ)h^ρ₂)du.

2. Inequalities via Riemann-Liouville fractional integrals In this section we find inequalities using Riemann-Liouville fractional integrals.

Theorem 2.1. Consider a differentiable mapping F : H → R on H^◦ and h₁, h₂ ∈ H with h₁ < h₂ and F⁰ ∈ L₁[h₁, h₂]. If |F⁰|^q, for q ≥ 1, is

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exponentially convex on [h1, h2], then the following inequality holds:

F (h₁) + F (h₂)

2 − Γ(α + 1)

2(h2− h₁)^αJ_h^α₁₊F (h^ρ₂) + J_h^α₂₋F (h₁)

≤ h₂− h₁ 2¹^q(α + 1)

1 − 1

2^α

F⁰(h₁) e^βh¹

q

+

F⁰(h₂) e^βh²

q¹_q . (6)

Proof. Suppose q = 1. Using Lemma 1.5 and exponential convexity of |F⁰|, we get

F (h₁) + F (h₂)

2 − Γ(α + 1)

2(h₂− h₁)^αJ_h^α₁₊F (h^ρ₂) + J_h^α₂₋F (h₁)

=

h₂− h₁ 2

Z 1 0

[(1 − u)^α− u^α] F⁰(uh₁+ (1 − u)h₂)du

≤ h₂− h₁ 2

Z 1 0

|(1 − u)^α− u^α||F⁰(uh₁+ (1 − u)h₂)|du

≤ h2− h₁ 2

Z 1 0

|(1 − u)^α− u^α|

u

F⁰(h1) e^βh¹

+ (1 − u)

F⁰(h2) e^βh²

=h₂− h₁ 2

Z 1/2 0

[(1 − u)^α− u^α]

u

F⁰(h₁) e^βh¹

+ (1 − u)

F⁰(h₂) e^βh²

+ Z 1

1/2

[u^α− (1 − u)^α]

u

F⁰(h₁) e^βh¹

+ (1 − u)

F⁰(h₂) e^βh²

. (7)

Since Z 1/2

0

u(1−u)^αdu−

Z 1/2 0

u^α+1du = Z 1

1/2

u^α(1−u)du−

Z 1 1/2

(1−u)^α+1du

= 1

(α + 1)(α + 2)−

1 2^α+1

α + 1, (8)

and Z 1/2

0

(1−u)^α+1du−

Z 1/2 0

u^α(1−u)du = Z 1

1/2

u^α+1du − Z 1

1/2

u(1−u)^αdu

= 1

α + 2 −

1 2^α+1

α + 1. (9)

Thus by using (8) and (9) in (7), we get (6) for q = 1. Now let q > 1.

Using power mean inequality on Lemma 1.5 and exponential convexity

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of |F⁰|^q, we obtain

F (h₁) + F (h2)

2 − Γ(α + 1)

2(h₂− h₁)^αJ_h^α

1+F (h^ρ₂) + J_h^α₂₋F (h₁)

=

h2− h₁ 2

Z ₁

0

[(1 − u)^α− u^α] F⁰(uh₁+ (1 − u)h₂)du

≤ h₂− h₁ 2

Z 1 0

|(1 − u)^α− u^α||F⁰(uh₁+ (1 − u)h₂)|du

≤ h₂− h₁ 2

Z 1 0

|(1 − u)^α− u^α|du

1−¹_q

Z 1 0

|(1 − u)^α− u^α||F⁰(uh1+ (1 − u)h2)|^qdu

¹_q

= h2− h₁ 2¹^q(α + 1)

1 − 1

2^α

F⁰(h1) e^βh¹

q

+

F⁰(h2) e^βh²

q¹_q . (10)

Since Z ₁

0

|(1 − u)^α− u^α|du = Z _1/2

0

[(1 − u)^α− u^α]du + Z ₁

1/2

[u^α− (1 − u)^α]du

= 2

α + 1

1 − 1

2^α

.

This completes the proof.

Remark 2.2. In Theorem 2.1.

(1) By letting β = 0, then the inequality (6) for q = 1 becomes the inequality (3.5) of Theorem 3 in [25].

(2) By letting β = 0 and α = 1, then the inequality (6) with q = 1 becomes the inequality 2.3 of Theorem 2.2 in [4].

Now we prove inequality for exponentially s-convex function in second sense as follows:

Theorem 2.3. Consider a differentiable mapping F : H ⊆ (0, ∞) → R on H^◦ and h₁, h₂ ∈ H with h₁ < h₂ and F⁰ ∈ L₁[h₁, h₂]. If |F⁰|^q, for q ≥ 1, is exponentially s-convex in second sense on [h₁, h₂], then the

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following inequality holds:

F (h₁) + F (h₂)

2 − Γ(α + 1)

2(h₂− h₁)^α J_h^α₁₊F (h^ρ₂) + J_h^α₂₋F (h₁)

≤ h₂− h₁ 2(α + 1)

2 α + 1

1 − 1

2^α

1−¹_q

F⁰(h₁) e^βh¹

q

+

F⁰(h₂) e^βh²

q¹_q

×

B(1

2; s + 1, α + 1) − B(1

2; α + 1, s + 1) + 2^α+s− 1 2^α+s(α + s + 1)

. (11)

Where B_v is an incomplete beta function defined by B_v(h₁, h₂) = Rv

0 t^h¹⁻¹(1 − t)^h²⁻¹dt, v ∈ (0, 1).

Proof. Suppose q = 1. Using Lemma 1.5 and exponential s-convexity of |F⁰|, we get

F (h₁) + F (h₂)

2 − Γ(α + 1)

2(h2− h₁)^αJ_h^α₁₊F (h^ρ₂) + J_h^α₂₋F (h₁)

=

h₂− h₁ 2

Z 1 0

[(1 − u)^α− u^α] F⁰(uh1+ (1 − u)h2)du

≤ h2− h₁ 2

Z 1 0

|(1 − u)^α− u^α||F⁰(uh1+ (1 − u)h2)|du

≤ h₂− h₁ 2

Z 1 0

|(1 − u)^α− u^α|

u^s

F⁰(h₁) e^βh¹

+ (1 − u)^s

F⁰(h₂) e^βh²

= ≤ h₂− h₁ 2

Z 1/2 0

[(1 − u)^α− u^α]

u^s

F⁰(h₁) e^βh¹

+ (1 − u)^s

F⁰(h₂) e^βh²

+ Z 1

1/2

[u^α− (1 − u)^α]

u^s

F⁰(h1) e^βh¹

+ (1 − u)^s

F⁰(h2) e^βh²

. (12)

Since (13)

Z 1/2 0

u^s(1 − u)^αdu = Z 1

1/2

u^α(1 − u)^sdu = B(1

2; s + 1, α + 1),

(14)

Z 1/2 0

u^α(1 − u)^sdu = Z 1

1/2

u^s(1 − u)^αdu = B(1

2; α + 1, s + 1),

(15)

Z 1/2 0

u^α+sdu = Z 1

1/2

(1 − u)^α+sdu = 1

2^s+α+1(s + α + 1),

(7)

and (16)

Z 1/2 0

(1 − u)^α+sdu = Z 1

1/2

u^α+sdu = 1

s + α + 1 − 1

2^s+α+1(s + α + 1). Thus by using (13)∼(16) in (12), we get

F (h₁) + F (h2)

2 − Γ(α + 1)

2(h₂− h₁)^α J_h^α

1+F (h^ρ₂) + J_h^α₂₋F (h₁)

≤ h₂− h₁ 2

F⁰(h₁) e^βh¹

+

F⁰(h₂) e^βh²

×

B(1

2; s + 1, α + 1) − B(1

2; α + 1, s + 1) + 2^α+s− 1 2^α+s(α + s + 1)

. (17)

Now let q > 1. Using power mean inequality on Lemma 1.5 and exponential s-convexity of |F⁰|^q, we obtain

F (h₁) + F (h2)

2 − Γ(α + 1)

2(h2− h₁)^αJ_h^α

1+F (h^ρ₂) + J_h^α₂₋F (h₁)

=

h2− h₁ 2

Z 1 0

[(1 − u)^α− u^α] F⁰(uh1+ (1 − u)h2)du

≤ h2− h₁ 2

Z 1 0

|(1 − u)^α− u^α||F⁰(uh1+ (1 − u)h2)|du

≤ h₂− h₁ 2

Z 1 0

|(1 − u)^α− u^α|du

1−¹_q

Z 1 0

|(1 − u)^α− u^α||F⁰(uh₁+ (1 − u)h₂)|^qdu

¹_q

= h₂− h₁ 2(α + 1)

2 α + 1

1 − 1

2^α

1−¹_q

F⁰(h₁) e^βh¹

q

+

F⁰(h₂) e^βh²

q¹_q

×

B(1

2; s+1, α+1) − B(1

2; α+1, s+1)+ 2^α+s− 1 2^α+s(α+s+1)

. (18)

Since Z ₁

0

|(1 − u)^α− u^α|du = Z _1/2

0

[(1 − u)^α− u^α]du + Z ₁

1/2

[u^α− (1 − u)^α]du

= 2

α + 1

1 − 1

2^α

.

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This completes the proof.

Remark 2.4. In Theorem 2.3.

(1) By letting β = 0, then the inequality (11) becomes the inequality (2.5) of Theorem 4 in [26].

(2) By letting β = 0 and α = 1, then the inequality (11) with q = 1 becomes the inequality of Theorem 1 in [22].

3. Inequalities via Katugampola fractional integrals First we prove the result for exponentially convex functions.

Theorem 3.1. Let α > 0, ρ > 0. Consider a differentiable mapping F : [h^ρ₁, h^ρ₂] → R on (h^ρ₁, h^ρ₂) with h^ρ₁ < h^ρ₂ and F⁰ ∈ L₁[h^ρ₁, h^ρ₂]. If |F⁰| is exponentially convex on [h^ρ₁, h^ρ₂], then the following inequality holds:

F (h^ρ₁) + F (h^ρ₂)

2 −αρ^αΓ(α + 1) 2(h^ρ₂− h^ρ₁)^α

_ρ

I_h^α₁₊F (h^ρ₂) + ^ρI_h^α₂₋F (h^ρ₁)

≤ h^ρ₂− h^ρ₁ 2ρ(α + 1)

1 − 1

2^α

F⁰(h^ρ₁) e^βh^ρ¹

+

F⁰(h^ρ₂) e^βh^ρ²

. (19)

Proof. Using Lemma 1.7 and exponential convexity of |F⁰|, we get

F (h^ρ₁) + F (h^ρ₂)

2 −αρ^αΓ(α + 1) 2(h^ρ₂− h^ρ₁)^α

_ρ

I_h^α₁₊F (h^ρ₂) +^ρI_h^α₂₋F (h^ρ₁)

=

h^ρ₂− h^ρ₁ 2

Z 1 0

[(1 − u^ρ)^α− (u^ρ)^α] u^ρ−1F⁰(u^ρh^ρ₁+ (1 − u^ρ)h^ρ₂)du

≤ h^ρ₂− h^ρ₁ 2

Z 1 0

u^ρ−1|(1 − u^ρ)^α− (u^ρ)^α||F⁰(u^ρh^ρ₁+ (1 − u^ρ)h^ρ₂)|du

≤ h^ρ₂− h^ρ₁ 2

Z ₁

0

u^ρ−1|(1−u^ρ)^α−(u^ρ)^α|

u^ρ

+(1−u^ρ)

= h^ρ₂−h^ρ₁ 2

Z 1/√^ρ 2 0

u^ρ−1[(1−u^ρ)^α−u^ρα]

u^ρ

+(1−u^ρ)

+ Z 1

1/√^ρ 2

u^ρ−1[u^ρα− (1 − u^ρ)^α]

u^ρ

+ (1 − u^ρ)

. (20)

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Since

Z 1/√^ρ 2 0

u^ρ−1u^ρ(1 − u^ρ)^αdu − Z 1/√^ρ

2 0

u^ρ−1u^ρ(α+1)du

= Z 1

1/√^ρ 2

u^ρ−1u^ρα(1 − u^ρ)du − Z 1

1/√^ρ 2

u^ρ−1(1 − u^ρ)^α+1du

= 1 ρ

"

1

(α + 1)(α + 2) −

1 2^α+1

(α + 1)

# , (21)

and

Z 1/√^ρ 2 0

u^ρ−1(1 − u^ρ)^α+1du − Z 1/√^ρ

2 0

u^ρ−1u^ρα(1 − u^ρ)du

= Z 1

1/√^ρ 2

u^ρ−1u^ρ(α+1)du − Z 1

1/√^ρ 2

u^ρ−1u^ρ(1 − u^ρ)^αdu

= 1 ρ

"

1 (α + 2) −

1 2^α+1

(α + 1)

# . (22)

Thus by using (21) and (22) in (20), we get (19).

Remark 3.2. In Theorem 3.1. By letting β = 0, then the inequality (19) becomes the inequality (17) of Theorem 2.5 in [2].

Theorem 3.3. Let α > 0, ρ > 0. function F : [h^ρ₁, h^ρ₂] ⊆ (0, ∞) → R on (h^ρ₁, h^ρ₂) with h^ρ₁ < h^ρ₂ and F⁰ ∈ L₁[h^ρ₁, h^ρ₂]. If |F⁰| is exponentially s-convex in second sense on [h^ρ₁, h^ρ₂], then the following inequality holds:

F (h^ρ₁) + F (h^ρ₂)

2 −αρ^αΓ(α + 1) 2(h^ρ₂− h^ρ₁)^α

_ρ

I_h^α₁₊F (h^ρ₂) + ^ρI_h^α₂₋F (h^ρ₁)

≤ h^ρ₂− h^ρ₁ 2ρ(α + 1)

+

×

B(1

2; s + 1, α + 1) − B(1

2; α + 1, s + 1) + 2^α+s− 1 2^α+s(α + s + 1)

. (23)

Where Bv is an incomplete beta function defined by Bv(h1, h2) = Rv

0 t^h¹⁻¹(1 − t)^h²⁻¹dt, v ∈ (0, 1).

(10)

Proof. Using Lemma 1.7 and exponential s-convexity of |F⁰|, we get

F (h^ρ₁) + F (h^ρ₂)

2 −αρ^αΓ(α + 1) 2(h^ρ₂− h^ρ₁)^α

_ρ

I_h^α₁₊F (h^ρ₂) +^ρI_h^α₂₋F (h^ρ₁)

=

h^ρ₂− h^ρ₁ 2

Z 1 0

[(1 − u^ρ)^α− (u^ρ)^α] u^ρ−1F⁰(u^ρh^ρ₁+ (1 − u^ρ)h^ρ₂)du

≤ h^ρ₂− h^ρ₁ 2

Z 1 0

u^ρ−1|(1 − u^ρ)^α− (u^ρ)^α||F⁰(u^ρh^ρ₁+ (1 − u^ρ)h^ρ₂)|du

≤ h^ρ₂− h^ρ₁ 2

Z 1 0

u^ρ−1|(1−u^ρ)^α−(u^ρ)^α|

u^ρs

+(1−u^ρ)^s

= h^ρ₂−h^ρ₁ 2

Z 1/√^ρ 2 0

u^ρ−1[(1−u^ρ)^α−u^ρα]

u^ρs

+(1−u^ρ)^s

+ Z 1

1/√^ρ 2

u^ρ−1[u^ρα− (1 − u^ρ)^α]

u^ρs

+ (1 − u^ρ)^s

. (24)

Since

Z 1/√^ρ 2 0

u^ρ−1u^ρs(1 − u^ρ)^αdτ = Z 1

1/√^ρ 2

u^ρ−1u^ρα(1 − u^ρ)^sdu

= 1 ρB(1

2; s + 1, α + 1), (25)

Z 1/√^ρ 2 0

u^ρ−1u^ρα(1 − u^ρ)^sdτ = Z 1

1/√^ρ 2

u^ρ−1u^ρs(1 − u^ρ)^αdu

= 1 ρB(1

2; α + 1, s + 1), (26)

Z 1/√^ρ 2 0

u^ρ−1u^ρ(α+s)du = Z 1

1/√^ρ 2

u^ρ−1(1 − u^ρ)^α+sdu

= 1 ρ

1

2^s+α+1(s + α + 1), (27)

(11)

and

Z 1/√^ρ 2 0

u^ρ−1(1 − u^ρ)^α+sdu = Z 1

1/√^ρ 2

u^ρ−1u^ρ(α+s)du

= 1 ρ

1

s + α + 1 − 1

2^s+α+1(s + α + 1)

. (28)

Thus by using (25)∼(28) in (24), we get (23).

Corollary 3.4. In Theorem 3.3. By letting β = 0, we get

F (h^ρ₁) + F (h^ρ₂)

2 −αρ^αΓ(α + 1) 2(h^ρ₂− h^ρ₁)^α

_ρ

I_h^α₁₊F (h^ρ₂) + ^ρI_h^α₂₋F (h^ρ₁)

≤ h^ρ₂− h^ρ₁ 2ρ(α + 1)

F⁰(h^ρ₁) +

F⁰(h^ρ₂)

×

B(1

2; s + 1, α + 1) − B(1

2; α + 1, s + 1) + 2^α+s− 1 2^α+s(α + s + 1)

. (29)

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Naila Mehreen

School of Natural Sciences,

National University of Sciences and Technology, Sector H-12 Islamabad, Pakistan.

E-mail: nailamehreen@gmail.com Matloob Anwar

School of Natural Sciences,

National University of Sciences and Technology, Sector H-12 Islamabad, Pakistan.

E-mail: matloob t@yahoo.com